Chapter 11, Problem 9QP

Returns and Standard Deviations Consider the following information:

1. a. Your portfolio is invested 30 percent each in A and C, and 40 percent in B. What is the expected return of the portfolio?
2. b. What is the variance of this portfolio? The standard deviation?

Summary Introduction

To determine: The Expected Return on Portfolio.

Introduction: Expected Return is a process of estimating the profits and losses an investor earns through the expected rate of returns. Standard deviation is apportioned of distribution of a collection of figures from its mean.

Answer to Problem 9QP

Solution: The Expected Return on Portfolio is 11.17%

Explanation of Solution

Determine the Portfolio Return for each Stock

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Good}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.30\times 0.09\right)+\left(0.40\times 0.10\right)+\left(0.30\times 0.15\right)\right]\\ =\left[0.027+0.04+0.045\right]\\ =0.112or11.20\%\end{array}$

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Boom}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.30\times 0.24\right)+\left(0.40\times 0.45\right)+\left(0.30\times 0.33\right)\right]\\ =\left[0.072+0.18+0.099\right]\\ =0.351or35.10\%\end{array}$

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Poor}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.30\times 0.03\right)+\left(0.40\times \left(-0.10\right)\right)+\left(0.30\times \left(-0.05\right)\right)\right]\\ =\left[0.009+\left(-0.04\right)+\left(-0.015\right)\right]\\ =-0.046or-4.60\%\end{array}$

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Bust}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.30\times \left(-0.05\right)\right)+\left(0.40\times \left(-0.25\right)\right)+\left(0.30\times \left(-0.09\right)\right)\right]\\ =\left[-0.015+\left(-0.1\right)+\left(-0.027\right)\right]\\ =-0.142or-14.20\%\end{array}$

Therefore the Portfolio Return for Boom is 35.10%, Good is 11.20%, Poor is -4.60% and Bust is -14.20%.

Determine the Expected Return on Portfolio

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_{Rp}\right)=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{Boom}\times \mathrm{Re}tur{n}_{Boom}\right)+\left(\mathrm{Pr}obabilit{y}_{Good}\times \mathrm{Re}tur{n}_{Good}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Poor}\times \mathrm{Re}tur{n}_{Poor}\right)+\left(\mathrm{Pr}obabilit{y}_{Bust}\times \mathrm{Re}tur{n}_{Bust}\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(25\%\times 35.10\%\right)+\left(35\%\times 11.20\%\right)+\\ \left(30\%\times \left(-4.60\%\right)\right)+\left(5\%\times \left(-14.20\%\right)\right)\end{array}\right]\\ =\left[0.08775+0.0448+\left(-0.0138\right)+\left(-0.0071\right)\right]\\ =0.11165or11.17\%\end{array}$

Therefore the Expected Return on Portfolio is 11.17%.

Summary Introduction

To determine: The Variance and Standard Deviation on Portfolio.

Answer to Problem 9QP

Solution: The Variance on Portfolio is 0.025 and Standard Deviation on Portfolio is 15.81%.

Explanation of Solution

Determine the Variance of Portfolio

$\begin{array}{c}Variance\left({\sigma }_P{}^2\right)=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{Boom}\times {\left(\mathrm{Re}tur{n}_{Boom}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Good}\times {\left(\mathrm{Re}tur{n}_{Good}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Poor}\times {\left(\mathrm{Re}tur{n}_{Poor}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Bust}\times {\left(\mathrm{Re}tur{n}_{Bust}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)+\end{array}\right]\\ =\left[\begin{array}{l}\left(25\%\times {\left(35.10\%-11.17\%\right)}^2\right)+\left(40\%\times {\left(11.20\%-11.17\%\right)}^2\right)+\\ \left(30\%\times {\left(-4.60\%-11.17\%\right)}^2\right)+\left(5\%\times {\left(-14.20\%-11.17\%\right)}^2\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(25\%\times 0.0572884\right)+\left(40\%\times 0.00000012\right)+\left(30\%\times 0.0248535\right)+\\ \left(5\%\times 0.0643383\right)\end{array}\right]\\ =\left[0.014322+0.000000049+0.00745606+0.00321692\right]\\ =0.02499513or0.025\end{array}$

Therefore the Variance on Portfolio is 0.025.

Determine the Standard Deviation of Portfolio

$\begin{array}{c}S\tan dardDeviation\left(\sigma \right)=\sqrt{Variance\left({\sigma }_P{}^2\right)}\\ =\sqrt{0.025}\\ =0.15811or15.81\%\end{array}$

Therefore the Standard Deviation on Portfolio is 15.81%.